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5.5: Entropy

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In addition to learning that the efficiency of a Carnot engine depends only on the high and low temperatures, more interesting things can be derived through the exploration of this system. For example, consider the total heat transferred in the cycle:

qtot=nRThln(V2V1)nRTlln(V4V3)

Making the substitution

V2V1=V3V4

the total heat flow can be seen to be given by

qtot=nRThln(V4V3)nRTlln(V4V3)

It is clear that the two terms do not have the same magnitude, unless Th=Tl. This is sufficient to show that q is not a state function, since it’s net change around a closed cycle is not zero (as any value of a state function must be.) However, consider what happens when the sum of q/T is considered:

qT=nRThln(V4V3)ThnRTlln(V4V3)Tl=nRln(V4V3)nRln(V4V3)=0

This is the behavior expected for a state function! It leads to the definition of entropy in differential form,

dSdqrevT

In general, dqrev will be larger than dq (since the reversible pathway defines the maximum heat flow.) So, it is easy to calculate entropy changes, as one needs only to define a reversible pathway that connects the initial and final states, and then integrate dq/T over that pathway. And since \(\Delta S\) is defined using q for a reversible pathway, ΔS is independent of the actual path a system follows to undergo a change.

Contributors and Attributions

  • Patrick E. Fleming (Department of Chemistry and Biochemistry; California State University, East Bay)


This page titled 5.5: Entropy is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Patrick Fleming.

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